Locally Connected Networks (\(\mathsf {LCN}_n\) ) [1]}, [2]} are convolutional networks without weight sharing between spatial locations. They share the connectivity pattern and topology of a standard convolutional network, but the weights are not shared across spatial patches. Mathematically, the network is defined as in Equation REF , but with all the shared parameters \(\omega _{ij, \beta }^{l}\)  replaced by unshared \(\omega _{ij, \alpha , \beta }^{l}\sim \mathcal {N}(0, 1)\) . In this work, we assume that the \(\mathsf {LCN}_n\)  are always associated with a vectorization readout layer and it is therefore clear that the function space of \(\mathsf {LCN}_n\)  is a superset of \(\mathsf {VEC}_n\) . Interestingly, \(\mathsf {LCN}_n\)  is also a subset of \(\mathsf {VEC}_{dn}\)  when the network is \(d\)  times wider.
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